Let r =2^d-1 ＋ 1. We investigate the diophantine inequality｜∑i=1^r λiФi（xi,yi）＋η｜〈（max 1≤i≤r{｜xi｜,｜yi｜}）^-δ,where Фi（x,y）∈X[x,y]（1≤i≤r） are nondegenerate forms of degree d = 3 or 4.
Let d ≥ 3 be an integer, and set r = 2^d-1 ＋ 1 for 3 ≤ d ≤ 4, r = 17 5~ ＂2441 for 5 ≤ d ≤ 6, r = d^2＋d＋1 for 7 ≤ d ≤ 8, and r = d^2＋d＋2 for d ≥ 9, respectively. Suppose that Фi（x, y） E Z[x, y] （1 ≤...Let d ≥ 3 be an integer, and set r = 2^d-1 ＋ 1 for 3 ≤ d ≤ 4, r = 17 5~ ＂2441 for 5 ≤ d ≤ 6, r = d^2＋d＋1 for 7 ≤ d ≤ 8, and r = d^2＋d＋2 for d ≥ 9, respectively. Suppose that Фi（x, y） E Z[x, y] （1 ≤ i ≤ r） are homogeneous and nondegenerate binary forms of degree d. Suppose further that λ1, λ2,. ..., λr are nonzero real numbers with λ1/λ2 irrational, and λ1λ1（x1, y1） ＋ λ2q）2（x2, y2） ＋ ... ＋ ）,λrФr（xr, yr） is indefinite. Then for any given real η and σ with 0 〈 cr 〈 22-d, it is proved that the inequalityhas infinitely ｜r∑i=1λФi（xi,yi）＋η｜〈（max 1≤i≤r{｜xi｜,｜yi｜}）^-σmany solutions in integers Xl, x2,..., xr, Yl, Y2,.--, Yr. This result constitutes an improvement upon that of B. Q. Xue.展开更多
基金Acknowledgements The author was grateful to his supervisor, Professor Hongze Li, for his guidance and support. The author would like to thank Quanwu Mu for his warm heart.He gave talks on diophantine inequalities to the author individually and provided helpful discussion. This work was supported by the National Natural Science Foundation of China （Grant No. 11271249） and Specialized Research Fund for the Doctoral Program of Higher Education （No. 20120073110059）.
文摘Let r =2^d-1 ＋ 1. We investigate the diophantine inequality｜∑i=1^r λiФi（xi,yi）＋η｜〈（max 1≤i≤r{｜xi｜,｜yi｜}）^-δ,where Фi（x,y）∈X[x,y]（1≤i≤r） are nondegenerate forms of degree d = 3 or 4.
文摘Let d ≥ 3 be an integer, and set r = 2^d-1 ＋ 1 for 3 ≤ d ≤ 4, r = 17 5~ ＂2441 for 5 ≤ d ≤ 6, r = d^2＋d＋1 for 7 ≤ d ≤ 8, and r = d^2＋d＋2 for d ≥ 9, respectively. Suppose that Фi（x, y） E Z[x, y] （1 ≤ i ≤ r） are homogeneous and nondegenerate binary forms of degree d. Suppose further that λ1, λ2,. ..., λr are nonzero real numbers with λ1/λ2 irrational, and λ1λ1（x1, y1） ＋ λ2q）2（x2, y2） ＋ ... ＋ ）,λrФr（xr, yr） is indefinite. Then for any given real η and σ with 0 〈 cr 〈 22-d, it is proved that the inequalityhas infinitely ｜r∑i=1λФi（xi,yi）＋η｜〈（max 1≤i≤r{｜xi｜,｜yi｜}）^-σmany solutions in integers Xl, x2,..., xr, Yl, Y2,.--, Yr. This result constitutes an improvement upon that of B. Q. Xue.
